Proper velocity

In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured displacement vector ${\displaystyle {\textbf {x}}}$ and proper time τ elapsed on the clocks of the traveling object:

${\displaystyle {\textbf {w}}={\frac {d{\textbf {x}}}{d\tau }}}$

It is an alternative to ordinary velocity, the distance per unit time where both distance and time are measured by the observer.

The two types of velocity, ordinary and proper, are very nearly equal at low speeds. However, at high speeds proper velocity retains many of the properties that velocity loses in relativity compared with Newtonian theory. For example, proper velocity equals momentum per unit mass at any speed, and therefore has no upper limit. At high speeds, as shown in the figure at right, it is proportional to an object's energy as well.

Proper velocity w can be related to the ordinary velocity v via the Lorentz factor γ:

${\displaystyle {\textbf {w}}={\frac {d{\textbf {x}}}{dt}}{\frac {dt}{d\tau }}={\textbf {v}}\,\gamma (v)}$

where t is coordinate time or "map time". For unidirectional motion, each of these is also simply related to a traveling object's hyperbolic velocity angle or rapidity η by

${\displaystyle \eta =\sinh ^{-1}{\frac {w}{c}}=\tanh ^{-1}{\frac {v}{c}}=\pm \cosh ^{-1}\gamma }$.